Exercise · ZW-MOBIUS-INVERSION-FORMULA-008 · ★★☆ Intermediate
If \(g(n)=\sum_{d\mid n}f(d)\) and \(f(n)=n\) for all \(n\), compute \(g(4)\).
Choose one option.
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- D. 7
Explanation
The divisors of 4 are 1, 2, and 4, so \(g(4)=1+2+4=7\). Leaving out any divisor changes the result.
Review the concept: Möbius inversion formula
Formula that recovers an arithmetic function \(f\) from its sum over divisors, using the Möbius function \(\mu\).
\[f(n)=\sum_{d\mid n}\mu(d)\,g\!\left(\frac{n}{d}\right)\]
More practice problems: möbius inversion formula
- ZW-MOBIUS-INVERSION-FORMULA-001 Which arithmetic function appears as the coefficient in the Möbius inversion formula?
- ZW-MOBIUS-INVERSION-FORMULA-002 What is the correct way to invert g(n)=∑_d| nf(d)?
- ZW-MOBIUS-INVERSION-FORMULA-003 Over which set is the sum taken in g(n)=∑_d| nf(d)?
- ZW-MOBIUS-INVERSION-FORMULA-004 If g(n)=∑_d| nμ(d) for every n, what is g(12)?
- ZW-MOBIUS-INVERSION-FORMULA-005 What is μ(4)?
- ZW-MOBIUS-INVERSION-FORMULA-006 If g(n)=n for all n, compute f(6) using Möbius inversion.
- ZW-MOBIUS-INVERSION-FORMULA-007 If g(n)=n^2 for all n, compute f(3) using Möbius inversion.
- ZW-MOBIUS-INVERSION-FORMULA-009 Let g(n)=n for all n. What is f(12) by Möbius inversion?
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